I'm coming from the assumption that you don't leave any backdoor open in either of the methods, and choose high-entropy passwords for Basic Auth. In that case, the biggest difference between the methods is that HTTPS will encrypt the traffic and prevent Man-in-the-Middle Attacks, making it impossible for anyone to eavesdrop, whereas Basic Auth will not.
So in that sense, the HTTPS based method is more secure.
You could achieve the same level of security for Basic Auth by adding HTTPS with a certificate that is commonly trusted by browsers (not requiring client-side authentication apart from Basic Auth)
In practical terms, you don't have to have the same password for everyone in the company. I think this is the greatest security flaw apart from any method, if security is a concern. Instead, you could have individual accounts for each person. If a person leaves, you only have to invalidate that account. It also gives you the ability to detect who, and from where, uses certain accounts, and if an account is identified as compromised, you can close that account without affecting others.